\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{a5paper}                   % ... or a4paper or a5paper or ... 
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\usepackage{MnSymbol}
\usepackage{tikz}
\usetikzlibrary{lindenmayersystems}
\usetikzlibrary{decorations.pathmorphing}
\usepackage{color}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}
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\begin{document}
\section{Introduction \& Objective }
\begin{itemize}
\item Some basic theory goes here goes here + why we do this

\item Lecture notes 4 goes thru a lot of it all.

\item plots of skeleton objects vs original objects as described on meeting after presentation

\item plots with various different types of objects (I can arrang code for all the described objects in the end of this file)
\end{itemize}

\section{One Scatterer}
\begin{itemize}
\item Some theory
\item Exercise 1, try different contours, different $M, N$, and different wave numbers $k$. What determines the numerical rank? We haven't really looked at different contours. See end of document for a couple of possible contours.
\item Exercise 2, how does the spectrum of $\mathbf{P}$ compare with the spectrum of $\mathbf{A}^{-1}$.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Two Scatterers}
\begin{itemize}
\item Some theory
\item Exercise 3, do exercise 1 and 2 again for two scatterers
\item Exercise 4, Create a combined $\mathbf{P}$
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Multiple Scatterers}
\begin{itemize}
\item Some theory
\item Exercise 5, Create a combined $\mathbf{P}$
\item Exercise 6, extend to arbitrary number of scatterers
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Closely Situated Scatterers}
\begin{itemize}
\item Exercise 7, handle closely situated scatterers
\item Exercise 8, what determines the mathematical properties of $S$ for a complicated domain.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Possible Extensions}
\begin{itemize}
\item Merge order-list should be easy, and then support odd number of objects
\item Variable $M, N$ for each scatterer, would be useful for different types of objects, just add two lists
\item Corners (could cutting the domain into pieces, just like for scatterers too close to each other, work? i.e. cut thru the corner to avoid the undefined normal, assume could be tested pretty easily with your code xavier) I have just watched a little in the Bremer article how they do it.
\item mex, compile, run a big example, 
\end{itemize}

\section{Conclusions}
Stuff goes here


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exercises}
\subsection*{1}
Let $\Gamma$ define a simple smooth contour in the plane and consider an exterior scattering problem as described
in XXX 4. Determine teh scattering matrix $\mathbf{S}$ associated with $\Gamma$, where $\mathbf{S}$ is defined by 
\[
\mathbf{S}=\mathbf{B}\mathbf{A}^{-1}\mathbf{C}.
\]
Compute the singular values of $\mathbf{S}$ for a few different contours, for different number of discretisations points ($N,M$) and for different wave numbers $k$. Form hypotheses for what deterines the numerical rank.
\section*{2}
Consider the situation where a modest number of discretization points are used to represent $\Gamma$ and $\partial D$. Compute the proxy matrix $\mathbf{P}$ as defined  as
\[
\mathbf{P}=\mathbf{U}^*\mathbf{A}^{-1}\mathbf{U}
\]
using brute force (e.g. using \texttt{id\_decomp.m}). How does the spectrum of $\mathbf{P}$ compare to the spectrum of $\mathbf{A}^{-1}$? 

\section*{3}
Repeat Exercises 1 and 2 for a situation with \textit{two} scatterers $\Gamma_1$ and $\Gamma_2$.

\section*{4}
Consider again the situation in Exercise 3 where you have two contours $\Gamma_1$ and $\Gamma_2$. Compute the proxy matrices $\mathbf{P}_1$ and $\mathbf{P}_2$ for eacg scatterer individually. Use this information to form the global proxy matrix $\mathbf{P}$ that represents both scatterers.

\section*{5} Now consider a situation with 4 scatterers $\{\Gamma_i\}_{i=4}^7$. Combine these hierarchically:

\[
\begin{array}{rl}
\Gamma_1=&\Gamma_4\cup \Gamma_5\cup\Gamma_6\cup\Gamma_7\\
\Gamma_2=&\Gamma_4\cup\Gamma_5\\
\Gamma_3=&\Gamma_6\cup\Gamma_7\\
\end{array}
\]

Compute the four proxy matrices $\{\mathbf{P}_i\}_{i=4}^7$ that represent each individual scatterer. Then construct the global proxy matrix $\mathbf{P}=\mathbf{P}_1$ via three steps:
\begin{itemize}
	\item Compute the proxy matrix $\mathbf{P}_2$ by ``merging'' the proxy matrices $\mathbf{P}_4$ and $\mathbf{P}_5$.
	\item Compute the proxy matrix $\mathbf{P}_3$ by ``merging'' the proxy matrices $\mathbf{P}_6$ and $\mathbf{P}_7$.
	\item Compute the proxy matrix $\mathbf{P}_1$ by ``merging'' the proxy matrices $\mathbf{P}_2$ and $\mathbf{P}_3$.
\end{itemize}

\section*{6}
Extend the hierarchical merging technique that you developed in Exercise 5 to the situation involving a whole cluster of scatterers.

\section*{7}
So far, we have assumed that all scatterers are sufficiently well-separated that each can be contained in a disc $D_i$ that contains only $\Gamma_i$. Now suppose the scatterers are closely packed, as shown in figure XXX. Can you modify the merging scheme so that we can do away with the condition that each scatterer is alone in its associated disc?

\section*{8}
Now that you have a code that can compute the scattering matrix associated with complicated aggregations of scatterers, revisit the problem you addressed in Exercise 1. What determines the mathematical properties of the proxy matrix $\mathbf{S}$ associated with a complicated scatterer?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{pics}
% soft star

Soft Star

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:2*pi,samples=100] plot ({1.5*cos(\x r) + (0.3/2)*cos((5+1)*\x r)+(0.3/2)*cos((5-1)*\x r)},{sin(\x r) + (0.3/2)*sin((5+1)*\x r) - (0.3/2)*sin((5-1)*\x r)});
%\node at (-0.4,1.0) {$\Gamma$};
\end{tikzpicture}

% half moon 
Half Moon

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:360,samples=100] plot ({cos(\x)*(0.5+cos(\x))},{sin(\x)});
\end{tikzpicture}
% dx: -sin(x)*(0.5+cos(x))+cos(x)*(-sin(x))=-sin(x)*(0.5+2*cos(x))
% dy: cos(x)

% peanut 1
Peanut 1

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:360,samples=100] plot ({cos(\x)*(0.5+2*cos(\x)*cos(\x))},{sin(\x)*(0.5+2*cos(\x)*cos(\x))});
\end{tikzpicture}
% x: cos(t)*(0.5+2cos^2(t))
% y: sin(t)*(0.5+2cos^2(t))
% dx: -sin(t)*(0.5+2cos^2(t))+cos(t)*(-4cos(t)sin(t))=-sin(t)*(0.5-2cos^2(t))
% dy: cos(t)*(0.5+2cos^2(t))+sin(t)*(-4cos(t)sin(t))=0.5+2*cos^2(t)-4cos(t)sin^2(t)=2.5-(4cos(t)+2)sin^2(t)

%peanut  http://www.sciencedirect.com/science/article/pii/S0955799712000185
Peanut 2 (corners)

www.sciencedirect.com/science/article/pii/S0955799712000185

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:360,samples=100] plot ({2*cos(\x)*sqrt(abs(0.125*cos(2*\x)+sqrt(abs(0.353*0.353*0.353*0.353-0.125*0.125*sin(2*\x)*sin(2*\x)))))},{2*sin(\x)*sqrt(abs(0.125*cos(2*\x)+sqrt(abs(0.353*0.353*0.353*0.353-0.125*0.125*sin(2*\x)*sin(2*\x)))))});
\end{tikzpicture}

% clover from bremer
Clover (corners)

www.math.ucdavis.edu/\~{ }bremer/scattering.pdf

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:360,samples=100] plot ({cos(\x)*(0.5+abs(cos(2*\x)))},{sin(\x)*(0.5+abs(cos(2*\x)))});
\end{tikzpicture}


% soft many pointed star
Soft Star 2

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:2*pi,samples=100] plot ({1.5*cos(\x r) + (0.3/2)*cos((10+1)*\x r)+(0.3/2)*cos((10-1)*\x r)},{sin(\x r) + (0.3/2)*sin((10+1)*\x r) - (0.3/2)*sin((10-1)*\x r)});
\end{tikzpicture}


% Ellipsoid
Ellipsoid

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:2*pi,samples=100] plot ({2*cos(\x r)},{sin(\x r)});
\end{tikzpicture}

% Snake
Snake (short)

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0.5*pi:3.5*pi,samples=100] plot ({\x-3},{1+sin(\x r)});
\draw [ultra thick,domain=0.5*pi:3.5*pi,samples=100] plot ({\x-3},{sin(\x r)});
\draw [ultra thick,domain=0.5*pi:1.5*pi,samples=100] plot ({0.5*cos(\x r)+0.5*pi-3},{0.5*sin(\x r)+1.5});
\draw [ultra thick,domain=1.5*pi:2.5*pi,samples=100] plot ({0.5*cos(\x r)+3.5*pi-3},{0.5*sin(\x r)-0.5});
\end{tikzpicture}


% Snake
Snake (long)

\begin{tikzpicture}[scale=0.3]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0.5*pi:8.5*pi,samples=100] plot ({\x-6},{1+sin(\x r)});
\draw [ultra thick,domain=0.5*pi:8.5*pi,samples=100] plot ({\x-6},{sin(\x r)});
\draw [ultra thick,domain=0.5*pi:1.5*pi,samples=100] plot ({0.5*cos(\x r)+0.5*pi-6},{0.5*sin(\x r)+1.5});
\draw [ultra thick,domain=1.5*pi:2.5*pi,samples=100] plot ({0.5*cos(\x r)+8.5*pi-6},{0.5*sin(\x r)+1.5});
\end{tikzpicture}

% half moon 
Half Moon and Half Moon rotated $0.75\pi$ radians

\begin{tikzpicture}[scale=0.6]
\path[use as bounding box] (-5,-5) rectangle (5,5);
\draw [ultra thick,domain=0:360,samples=100] plot ({cos(\x)*(0.5+cos(\x))-4},{sin(\x)});

\draw [ultra thick,domain=0:2*pi,samples=100] plot ({cos(3*pi/4 r)*cos(\x r)*(0.5+cos(\x r))-sin(3*pi/4 r)*sin(\x r)},{sin(3*pi/4 r)*cos(\x r)*(0.5+cos(\x r))+cos(3*pi/4 r)*sin(\x r)});
\end{tikzpicture}


\end{document}
